We study coherent and incoherent diffractive hadron production in high-energy quarkonium–heavy nucleus collisions as a probe of the gluon saturation regime of quantum chromodynamics. Taking this process as a model for pA collisions, we argue that the coherent diffractive gluon production, in which the target nucleus stays intact, exhibits a remarkable sensitivity to the energy, rapidity, and atomic number dependence. The incoherent diffractive gluon production is less sensitive to the details of the low-x dynamics but can serve as a probe of fluctuations in the color glass condensate. As a quantitative measure of the nuclear effects on diffractive hadron production we introduce a new observable—the diffractive nuclear modification factor. We discuss possible signatures of gluon saturation in diffractive gluon production at the Relativistic Heavy Ion Collider, the Large Hadron Collider, and Electron Ion Collider.